Every proportional equation has a graph similar to this one. It is a straight line and goes through the point (0, 0). This is because if x = 0 then y must be equal to 0 as well. The constant of proportionality is what determines how steep the line is.

Examples

Example 1

Determine whether the following linear relationships represent proportional relationships:

y=2x+3

Worked Solution

Create a strategy

Proportional relationships can generally be written in the form: y = kx

Apply the idea

The given equation, y=2x+3, has a +3 term which makes it different from the proportional relationship equation. Therefore, this is not a proportional relationship.

y=0.75x

Worked Solution

Create a strategy

Proportional relationships can generally be written in the form: y = kx

Apply the idea

The given equation, y=0.75x, has no extra terms and resembles the proportional relationship equation. Therefore, this is a proportional relationship.

Idea summary

Proportional relationships can generally be written in the formy = kxwhere k is the constant of proportionality.

Represent proportions using equations

Not only can we determine if a relationship is proportional from an equation, but we can also write these equations to represent the relationship. We can use the known values of coordinate points from our proportional relationship to solve for k. For example, if we have a proportional relationship represented by the following graph:

We can use the given point, \left(2,1\right), in order to solve for k:

\displaystyle y	\displaystyle =	\displaystyle kx	Write the equation for proportional relationship
\displaystyle 1	\displaystyle =	\displaystyle k\times2	Substitute values of x and y
\displaystyle 1\div2	\displaystyle =	\displaystyle k\times 2 \div 2	Divide both sides by 2
\displaystyle \dfrac{1}{2}	\displaystyle =	\displaystyle k	Simplify
\displaystyle y	\displaystyle =	\displaystyle \dfrac{1}{2} \times x	Substitute the value of k

Therefore the equation that represents the proportional relationship is y=\dfrac{1}{2} x.

When provided with a graph, table, or situation representing a proportional relationship, it is important to note that any coordinate pair can be used. For instance, in the above graph, we could have used any point from the graph, such as \left(4,2\right).

However, we cannot use the point \left(0,0\right) because that is a shared point amongst all proportional relationships and does not help us solve for k.

Examples

Example 2

Consider the following table:

x	1	3	5	7
y	10	30	50	70

Determine whether the linear relationship represents a proportional relationship.

Worked Solution

Create a strategy

Proportional relationships can generally be written in the form: y = kx

Apply the idea

We can determine if the table of values representing linear relationshp also represents a proportional relationship by trying out at least two coordinates in the equation y=kx and see if we get the same value of k each time.

Let's start by trying out \left(1, 10\right).

\displaystyle 10	\displaystyle =	\displaystyle k \times 1	Substitute x=1 and y=10
\displaystyle 10	\displaystyle =	\displaystyle k	Simplify

Let's now try \left(3, 30\right).

\displaystyle 30	\displaystyle =	\displaystyle k \times 3	Substitute x=3 and y=30
\displaystyle 30\div3	\displaystyle =	\displaystyle k \times 3\div3	Divide both sides by 3
\displaystyle 10	\displaystyle =	\displaystyle k	Simplify

Let's try one more point, \left(5, 50\right), to confirm that k=10.

\displaystyle 50	\displaystyle =	\displaystyle k \times 5	Substitute x=5 and y=50
\displaystyle 50\div5	\displaystyle =	\displaystyle k \times 5\div5	Divide both sides by 5
\displaystyle 10	\displaystyle =	\displaystyle k	Simplify

Since we know that the relationship is linear and that the value of k is the same for three points, we can say that this is a proportional relationship. Therefore, this table of values represents a proportional relationship.

Determine the equation that represents the proportional relationship.

Worked Solution

Create a strategy

Proportional relationships can generally be written in the form: y=kx

Apply the idea

Since we know that k=10, we can write a proportional relationship to represent the table of values: y=10x

Example 3

Frank serves 2 cups of coffee every 4 minutes. Let y represent the number of cups of coffee, and x the number of minutes that have passed.

Write an equation where y is the subject that represents this proportional relationship.

Worked Solution

Create a strategy

Use the equation y = kx where k is the constant.

Apply the idea

Since y is the number of cups of coffee, and x is the amount of minutes that have passed, then y = 2 and x = 4.

\displaystyle y	\displaystyle =	\displaystyle kx	Write the equation for proportional relationship
\displaystyle 2	\displaystyle =	\displaystyle k \times 4	Substitute the given x and y-values
\displaystyle k	\displaystyle =	\displaystyle 2 \div 4	Divide 2 by 4 to get the value of k
\displaystyle k	\displaystyle =	\displaystyle \dfrac{2}{4}	Write in fraction form
\displaystyle k	\displaystyle =	\displaystyle \dfrac{1}{2}	Simplify

So, the proportional relationship is represented by: y = \dfrac{1}{2}x

Idea summary

We can write equations that represent proportional relationships by using known coordinate points to first solve for k, followed by substituting the value of k into the equation y=kx.

Outcomes

7.RP.A.2

Recognize and represent proportional relationships between quantities.

7.RP.A.2.B

Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.

7.RP.A.2.C

Represent proportional relationships by equations.

3.05 Equations representing proportional relationships

Ideas

Determine if a relationship is proportional

Examples

Example 1

Represent proportions using equations

Examples

Example 2

Example 3

Outcomes

7.RP.A.2

7.RP.A.2.B

7.RP.A.2.C

What is Mathspace

About Mathspace